Starter activity Can you make your calculator display the following sequences? Find the 20 th term for each of these sequences.
Can you find the next two terms of the following sequence 4, 8, 16, 32,....? Geometric Sequences
Geometric sequences Position number Sequence x2 4, 8, 16, 32,.... A sequence is geometric if where r is a constant called the common ratio x2
Geometric sequences or geometric progressions, hence the GP notation Different ways to describe this sequence: By listing its first few terms: 4, 8, 16, 32,... By specifying the first term and the common ratio: 1 st term is 4 and common ratio is 2 or By giving its nth term ? By graphical representation ?
Finding the nth term Position number 12345n Sequence x14x24x44x84x16 4x2 0 4x2 1 4x2 2 4x2 3 4x2 4 nth term = 4x2 n-1
4, 8, 16,... is a divergent sequence
Geometric sequences Can you find the next two terms of the following sequence? 0.2, 0.02, 0.002,.... Can you describe this sequence in different ways? By listing its terms: By specifying the first term and the common ratio: By finding its nth term: By graphical representation:
0.2, 0.02, 0.002,... is a convergent sequence The sequence converges to a certain value (or a limit number)
e.g. it approaches 0 This convergent sequence also oscillates. Another example of a convergent sequence:
Geometric sequences 1. Can you generate (or find) the first 5 terms of the following GPs? Seq A: Seq B: 2. Can you write down the nth term of these sequences? 3. Are these sequences convergent or divergent? Can you use the limit notation in your answers?
Geometric sequences 1. What is the ratio and the 7 th term for each of the following GPs? Seq A: 2, 10, 50, 250,...? Seq B: 24,12, 6, 3,....? Seq C: -27, 9, -3, 1,....? Challenge 1 What if you want to find the 50 th term of each of these sequences? How would you change your approach? Challenge 2 The 3 rd term in a geometric sequence is 36 and the 6 th term is 972. What is the value of the 1 st term and the common ratio? Challenge 3 Q6 handout
Suppose we have a 2 metre length of string which we cut in half We leave one half alone and cut the 2 nd in half again... and again cut the last piece in half Geometric Series
Continuing to cut the end piece in half, we would have in total In theory, we could continue for ever, but the total length would still be 2 metres, so This is an example of an infinite series.
or is the Greek capital letter S, used for Sum
Geometric series The sum of all the terms of a geometric sequence is called a geometric series. We can write the sum of the first n terms of a geometric series as: When n is large, how efficient is this method ? S n = a + ar + ar 2 + ar 3 + … + ar n –1 For example, the sum of the first 5 terms of the geometric series with first term 2 and common ratio 3 is: S 5 = 2 + (2 × 3) + (2 × 3 2 ) + (2 × 3 3 ) + (2 × 3 4 ) = = 242
The sum of a geometric series Start by writing the sum of the first n terms of a general geometric series with first term a and common ratio r as: Multiplying both sides by r gives: S n = a + ar + ar 2 + ar 3 + … + ar n –1 rS n = ar + ar 2 + ar 3 + … + ar n –1 + ar n Now if we subtract the first equation from the second we have: rS n – S n = ar n – a S n ( r – 1) = a ( r n – 1) Challenge: Can you follow the proof of the formula for the sum of the first n terms of a GS? (in pairs)
Geometric series a)Find the sum of the first 7 terms of the following GP: 4, - 2, 1,... giving your answer correct to 3 significant figures. b)Calculate: Challenge Is ? What is as an exact fraction?